In this article, the NRICH team describe the process of selecting solutions for publication on the site.

This article for primary teachers suggests ways in which to help children become better at working systematically.

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

How many solutions can you find to this sum? Each of the different letters stands for a different number.

We're excited about this new program for drawing beautiful mathematical designs. Can you work out how we made our first few pictures and, even better, share your most elegant solutions with us?

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

How many different triangles can you make on a circular pegboard that has nine pegs?

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

A package contains a set of resources designed to develop students’ mathematical thinking. This package places a particular emphasis on “being systematic” and is designed to meet. . . .

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

A challenging activity focusing on finding all possible ways of stacking rods.

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

Many natural systems appear to be in equilibrium until suddenly a critical point is reached, setting up a mudslide or an avalanche or an earthquake. In this project, students will use a simple. . . .

A Sudoku with clues given as sums of entries.

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

This challenge extends the Plants investigation so now four or more children are involved.

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

Can you find all the different triangles on these peg boards, and find their angles?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

How many trains can you make which are the same length as Matt's, using rods that are identical?

Try out the lottery that is played in a far-away land. What is the chance of winning?

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

A particular technique for solving Sudoku puzzles, known as "naked pair", is explained in this easy-to-read article.

Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?

Can you find all the different ways of lining up these Cuisenaire rods?

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

Find out what a "fault-free" rectangle is and try to make some of your own.

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

In this matching game, you have to decide how long different events take.

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?